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A361642
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Triangle read by rows where row n is a self-inverse permutation of 1..n formed starting from a column 1..n and sliding numbers to the right and down.
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3
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1, 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 3, 4, 2, 1, 6, 4, 3, 5, 2, 1, 7, 4, 3, 5, 6, 2, 1, 8, 5, 6, 3, 4, 7, 2, 1, 9, 5, 4, 3, 7, 6, 8, 2, 1, 10, 6, 4, 8, 3, 7, 5, 9, 2, 1, 11, 6, 8, 5, 3, 9, 4, 7, 10, 2, 1, 12, 7, 5, 4, 10, 3, 8, 9, 6, 11, 2, 1, 13, 7, 5, 4, 6, 3, 11, 10, 9, 8, 12, 2, 1, 14, 8, 10, 11, 6, 12, 3, 9, 4, 5, 7, 13, 2
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OFFSET
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1,3
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COMMENTS
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The transformation process starts with an upwards column of numbers 1..n.
The rightmost number of the topmost row slides right across columns 1 smaller (if any), and then drops down either onto a 2 or more smaller final column if there is one, or otherwise starting a new final column.
Sliding steps continue until reaching a single row (all columns height 1), which is triangle row n.
The final move is always with 2 that slides rightward then one step down to the end of the resulting single row. 1 never moves.
The resulting row is a self-inverse permutation (involution) because the reverse steps are exactly those performed if the row is stood up as the starting column again.
In the case of transforming the column of a composite n, there will be instances when a number sliding sideways will drop down just one step as the changing stack becomes a complete rectangle of rows and columns. Such a number is always greater by 1 than the height of the completed rectangle, whose height and width are divisors of n.
The travel of a sliding number, other than 2, that thus completes a rectangle will continue: it immediately moves sideways again one step, then downwards. Those n's that are prime numbers will not have such number since their changing stack can never form a rectangle with divisors greater than 1, except themselves.
When performing the process in an orthogonal grid where the numbers slide sideways and downward in discrete steps, the total steps for n is an oblong number, the sequence of which is A002378.
In the orthogonal grid, in mid-process, the bounding box of the changing stack of numbers follows the n = x*y curve, and is in exact contact with it at integer x and y points where n is composite.
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LINKS
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FORMULA
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EXAMPLE
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Triangle T(n,k) begins:
n/k | 1 2 3 4 5 6 7
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1 | 1;
2 | 1, 2;
3 | 1, 3, 2;
4 | 1, 4, 3, 2;
5 | 1, 5, 3, 4, 2;
6 | 1, 6, 4, 3, 5, 2;
7 | 1, 7, 4, 3, 5, 6, 2;
...
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A few snapshots of the process for n = 7, a prime number:
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7
6
5
4 4
3 3 5 3 5 3
2 2 6 2 6 2 6 5 2 6 5
1 1 7 1 7 4 1 7 4 1 7 4 3 1 7 4 3 5 6 2
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An example showing some stages of the process for a composite n = 6, with completed rectangles:
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6
5
4
3 3 4
2 2 5 2 5 3
1 1 6 1 6 4 1 6 4 3 5 2
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Step-by-step animation frames, showing 8, the rightmost number of the top row, sliding and dropping during its second movement, in the operation for n = 11:
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4 8 4 8 4 8 4 4 4
3 9 5 3 9 5 3 9 5 3 9 5 8 3 9 5 3 9 5
2 10 7 2 10 7 2 10 7 2 10 7 2 10 7 8 2 10 7
1 11 6 1 11 6 1 11 6 1 11 6 1 11 6 1 11 6 8
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PROG
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(MATLAB)
a = 1;
for r = 2:max_row
p = [1:r];
for k = 2:r-1
j = [1:r];
t1 = find(mod(j, k) == 0);
t2 = find(mod(j, k) ~= 0);
j(t1) = [r:-1:r-length(t1)+1];
j(t2) = [1:length(t2)];
p = p(j);
end
a = [a p];
end
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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